Condense Mixed Convexity and Optimization with an Application in Data Service Optimization

Abstract

Elements of matrix theory are useful in exploring solutions for optimization, data mining, and big data problems. In particular, mixed integer programming is widely used in data based optimization research that uses matrix theory (see for example [13]). Important elements of matrix theory, such as Hessian matrices, are well studied for continuous (see for example [11]) and discrete [9] functions, however matrix theory for functions with mixed (i.e. continuous and discrete) variables has not been extensively developed from a theoretical perspective. There are many mixed variable functions to be optimized that can make use of a Hessian matrix in various fields of research such as queueing theory, inventory systems, and telecommunication systems. In this work we introduce a mixed Hessian matrix, named condense mixed Hessian matrix, for mixed variable closed form functions $g: \mathbb {Z}^{n}\times \mathbb {R}^{m}\rightarrow \mathbb {R}$, and the use of this matrix for determining convexity and optimization results for mixed variable functions. These tasks are accomplished by building on the definition of a multivariable condense discrete convex function and the corresponding Hessian matrix that are introduced in [14]. In addition, theoretical condense mixed convexity and optimization results are obtained. The theoretical results are implemented on an M/M/s queueing function that is widely used in optimization, data mining, and big data problems.